> Even assuming that some set of automated probes have been travelling our galaxy for millions of years, and reached earth in the last few thousand years, where are they now? By what mechanism could they even be hiding? Every inch of the planet is being watched by satellites.

The most obvious way to hide is stay small. I don't know what physical limits would be to build small robots with advanced technology. Build main base somewhere in asteroid belt. Build some intermediate bases for refilling or something like that in moons. Or may be just rock somewhere in the space with small solar panels which collects electricity and can charge robot of size of sparrow which will travel further to Earth. Can we notice sparrow landing Earth and intelligently hiding e.g. in the sea?